Time-space hardness of learning sparse parities

Gillat Kol, Ran Raz, Avishay Tal

Research output: Chapter in Book/Report/Conference proceedingConference contribution

31 Scopus citations


We define a concept class F to be time-space hard (or memory-samples hard) if any learning algorithm for F requires either a memory of size super-linear in n or a number of samples super-polynomial in n, where n is the length of one sample. A recent work shows that the class of all parity functions is time-space hard [Raz, FOCS'16]. Building on [Raz, FOCS'16], we show that the class of all sparse parities of Hamming weight l is time-space hard, as long as l ≥ ω(log n/log log n). Consequently, linear-size DNF Formulas, linear-size Decision Trees and logarithmic-size Juntas are all time-space hard. Our result is more general and provides time-space lower bounds for learning any concept class of parity functions. We give applications of our results in the field of bounded-storage cryptography. For example, for every ω(log n) ≤ k ≤ n, we obtain an encryption scheme that requires a private key of length k, and time complexity of n per encryption/decryption of each bit, and is provably and unconditionally secure as long as the attacker uses at most o(nk) memory bits and the scheme is used at most 2o(k) times. Previously, this was known only for k = n [Raz, FOCS'16].

Original languageEnglish (US)
Title of host publicationSTOC 2017 - Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing
EditorsPierre McKenzie, Valerie King, Hamed Hatami
PublisherAssociation for Computing Machinery
Number of pages14
ISBN (Electronic)9781450345286
StatePublished - Jun 19 2017
Event49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017 - Montreal, Canada
Duration: Jun 19 2017Jun 23 2017

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
VolumePart F128415
ISSN (Print)0737-8017


Other49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017

All Science Journal Classification (ASJC) codes

  • Software


  • Bounded storage cryptography
  • Branching program
  • Fourier analysis
  • Lower bounds
  • PAC learning
  • Time-space tradeoff


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